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Theorem 1arymaptfv 47606
Description: The value of the mapping of unary (endo)functions. (Contributed by AV, 18-May-2024.)
Hypothesis
Ref Expression
1arymaptfv.h 𝐻 = ( ∈ (1-aryF 𝑋) ↦ (𝑥𝑋 ↦ (‘{⟨0, 𝑥⟩})))
Assertion
Ref Expression
1arymaptfv (𝐹 ∈ (1-aryF 𝑋) → (𝐻𝐹) = (𝑥𝑋 ↦ (𝐹‘{⟨0, 𝑥⟩})))
Distinct variable groups:   ,𝐹,𝑥   ,𝑋,𝑥
Allowed substitution hints:   𝐻(𝑥,)

Proof of Theorem 1arymaptfv
StepHypRef Expression
1 fveq1 6884 . . 3 ( = 𝐹 → (‘{⟨0, 𝑥⟩}) = (𝐹‘{⟨0, 𝑥⟩}))
21mpteq2dv 5243 . 2 ( = 𝐹 → (𝑥𝑋 ↦ (‘{⟨0, 𝑥⟩})) = (𝑥𝑋 ↦ (𝐹‘{⟨0, 𝑥⟩})))
3 1arymaptfv.h . 2 𝐻 = ( ∈ (1-aryF 𝑋) ↦ (𝑥𝑋 ↦ (‘{⟨0, 𝑥⟩})))
4 eqid 2726 . . . . 5 (0..^1) = (0..^1)
54naryrcl 47597 . . . 4 ( ∈ (1-aryF 𝑋) → (1 ∈ ℕ0𝑋 ∈ V))
65simprd 495 . . 3 ( ∈ (1-aryF 𝑋) → 𝑋 ∈ V)
76mptexd 7221 . 2 ( ∈ (1-aryF 𝑋) → (𝑥𝑋 ↦ (‘{⟨0, 𝑥⟩})) ∈ V)
82, 3, 7fvmpt3 6996 1 (𝐹 ∈ (1-aryF 𝑋) → (𝐻𝐹) = (𝑥𝑋 ↦ (𝐹‘{⟨0, 𝑥⟩})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3468  {csn 4623  cop 4629  cmpt 5224  cfv 6537  (class class class)co 7405  0cc0 11112  1c1 11113  0cn0 12476  ..^cfzo 13633  -aryF cnaryf 47592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-naryf 47593
This theorem is referenced by:  1arymaptf1  47608
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